Wave 0-trace and Length Spectrum on Convex Co-compact Hyperbolic Manifolds

Abstract

For convex co-compact hyperbolic quotients \H n+1 , we obtain a formula relating the 0-trace of the wave operator with the resonances and some conformal invariants of the boundary, generalizing a formula of Guillop and Zworski in dimension 2. Then, by writing this 0-trace with the length spectrum, we prove precise asymptotics of the number of closed geodesics with an effective, exponentially small error term when the dimension of the limit set of is greater than n/2.